Comparison Principles And Asymptotic Behavior Of Entropy Solutions Of PDE

In this thesis, some special phenomenon is investigated for first or second order partial di?erential equations. The main contents include the nonuniqueness due to the source functions’ singularity and the existing time of entropy solutions for first and second order evolution equations in the entire space; also the uniqueness of the asymptotic stationary solutions of the balance law on bounded intervals. Only nonnegative solutions are considered when dealing with problems of the entire space.For the autonomic balance law ut + ?· F(u) = f(u)in RN, suppose the flux function satisfies the condition of uniqueness: locally 1-1N-th order H ¨older continuity. If the source f(u) is not Lipschitz continuous near u = 0,then by linear approximation to it,two sequences of problems satisfying uniqueness condition are constructed. Taking limits brings out the maximal and minimal entropy solutions of the original balance law. At the same time, the comparison principle is extended and applied to distinguish whether there is blow-up phenomenon for kinds of entropy solutions. Examples are given to illustrate the conclusions above.For the parabolic equation ut- ?b(u) + ? · F(u) = f(u), the Cauchy problem in RNis investigated where b(·) is nondecreasing continuous, including strong degenerate case; while the flux and source functions are the same as the problems of balance law.By use of the linear approximation similar as the case of balance law, the maximal and minimal entropy solutions are obtained via constructing two sequences of problems satisfying uniqueness condition and taking limits in the frame of L∞(IRN×(0, T)).For the equationut +(F(u))x = f(x, u) with fixed boundary values on a bounded interval, the uniqueness conditions of asymptotic stationary entropy solution are investigated. When the flux is concave in certain sense, suppose the source has positive lower bound or negative upper bound, combining with the analysis on the boundary values, several su?cient conditions of uniqueness relative to the interval’s length are achieved. These are supplements to the already known conditions of uniqueness. Examples are also given to support the conclusions.